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Properties

Label 16.0.19385168517...9601.5
Degree $16$
Signature $[0, 8]$
Discriminant $37^{14}\cdot 173^{14}$
Root discriminant $2140.26$
Ramified primes $37, 173$
Class number $60553827504$ (GRH)
Class group $[6, 6, 1682050764]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![33380320760954558557, 2427071999215910603, -73879184588145834, -18683145716368537, -764954674399913, -77428720044109, 941787543275, 450226956190, 5083665532, 192556636, 51739983, -5352141, 47690, 2531, 594, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 594*x^14 + 2531*x^13 + 47690*x^12 - 5352141*x^11 + 51739983*x^10 + 192556636*x^9 + 5083665532*x^8 + 450226956190*x^7 + 941787543275*x^6 - 77428720044109*x^5 - 764954674399913*x^4 - 18683145716368537*x^3 - 73879184588145834*x^2 + 2427071999215910603*x + 33380320760954558557)
 
gp: K = bnfinit(x^16 - x^15 + 594*x^14 + 2531*x^13 + 47690*x^12 - 5352141*x^11 + 51739983*x^10 + 192556636*x^9 + 5083665532*x^8 + 450226956190*x^7 + 941787543275*x^6 - 77428720044109*x^5 - 764954674399913*x^4 - 18683145716368537*x^3 - 73879184588145834*x^2 + 2427071999215910603*x + 33380320760954558557, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 594 x^{14} + 2531 x^{13} + 47690 x^{12} - 5352141 x^{11} + 51739983 x^{10} + 192556636 x^{9} + 5083665532 x^{8} + 450226956190 x^{7} + 941787543275 x^{6} - 77428720044109 x^{5} - 764954674399913 x^{4} - 18683145716368537 x^{3} - 73879184588145834 x^{2} + 2427071999215910603 x + 33380320760954558557 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(193851685179255766530952003996759265201819225743449601=37^{14}\cdot 173^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2140.26$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $37, 173$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{15} + \frac{2063882011275354549939045611256051835715997755541675155595346126529247282107334546364945437827156295754692687277962706458118421855279}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{14} + \frac{4886649261881321654891222110603339106747873313881745126582800306900471715295504317026582969658419754563022488639299972694817334002137}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{13} + \frac{3557976590031371658548119640657478789600456742516738027770181176905466608322132014881680179935609662285949456676104853098237652024826}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{12} - \frac{406968640052887294830562711051449808305104154587440253803232954556302024491475247535281868550010735913232925106600581826328286741358}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{11} + \frac{224077224026087766267392324972104386337876200956488014393853184625489820040897805757546745080387594835007379018230413462590731523185}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{10} - \frac{3312910262618490935374743448371241279444990579624735157312650558880699808549871017516175406829165447521392356388301973079601294087057}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{9} + \frac{3115539155517868911756677093227016732054516266195839164248150448847872842215677788015084011554871361089673818811641712046576648580759}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{8} + \frac{3437949395726020439741610823539778554602016293454695659254410282476529312066916910053171325467124389285534756046510123831158333158694}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{7} + \frac{2692211581397263080746045840849025793838725786740859711736117854748139168512148070891545954533399134113153318409560868702913878227841}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{6} - \frac{122933564541432763864011196387753329364566474975724550826623889587512833528373329878973236669399499588095431756963597715043729845278}{275663842582036206202045334346030856107511905373238687083474347100483900644871160179200090332490492638185660507126469777425902092497} a^{5} - \frac{3890378825447773147229035029310865404674365770600863017690371479885076562246618512099441565626586551901451990238776612464433023062559}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{4} - \frac{5903788897826673503378462929550167737713429195724374533998803201938571668839150216538463297926562020757361548669655549574308188076259}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{3} + \frac{2912123210432428108013390707032532118893842367281698642390624746340786334769366178821740445740830439737197091673679582229147976376973}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a^{2} - \frac{1905904095163938529380575670685364338240427589938332625956617454326984113448636034372530825903968023174581670489962355549780544013118}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359} a - \frac{2818709369232428932639212448110350575421008625923152396634939016980600857804066457088638663616891345201297105548042345118889804844352}{12956200601355701691496130714263450237053059552542218292923294313722743330308944528422404245627053153994726043834944079539017398347359}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{6}\times C_{6}\times C_{1682050764}$, which has order $60553827504$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1189759845.59 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{173}) \), \(\Q(\sqrt{6401}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{37}, \sqrt{173})\), 8.8.68783926416507494438401.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$173$173.8.7.2$x^{8} - 692$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
173.8.7.2$x^{8} - 692$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$